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 BURP: 
 A BUNDLE REPEATER AND RESTORER 
 
 by 
 
 Bernard Ka Pang Tse 
 
 December, 197^ 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 IHE LIBRARY OF THE 
 
 FEB&<! 1975 
 
 UNIVERSITY OF ILLINOIS 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/burpbundlerepeat689tseb 
 
UIUCDCS-R-TU-689 
 
 BURP: 
 A BUNDLE REPEATER AND RESTORER 
 
 BY 
 
 BERNARD KA PANG TSE 
 
 December, 197^ 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 This work was supported in part by Contract No. US Navy 0001U-6T-A-0305-002U 
 and was submitted in partial fulfillment of the requirements for the degree 
 of Master of Science in Electrical Engineering at the University of Illinois, 
 
Ill 
 
 ACKNOWLEDGEMENT 
 
 The author is much indebted to Professor W. J. Poppelbaum, Director 
 of the Information Engineering Laboratory, who suggested the thesis topic 
 and who, through the years, has provided invaluable counsel, understanding 
 and support. His help and friendship are highly appreciated. 
 
 The author is also very grateful to Professor Chushin Afuso (now of 
 Yamagata University, Japan) who, while he was at the University of Illi- 
 nois , provided the first year graduate student with much guidance and 
 insight, giving a lot of his valuable time through countless afternoons! 
 
 Thanks are also due to Mr. Frank Serio and the Electronic Fabrication 
 Group, whose efforts made the construction of the BURP System possible, 
 and to Ms. Evelyn Huxhold, who typed this thesis. 
 
IV 
 
 PREFACE 
 
 This thesis examines the problem of broken wires in bundles: It is 
 shown that a much better approximation to the original information can 
 be obtained by connecting the ends of broken wires to an arbitrary neigh- 
 bor in a so-called "Repeater." Further investigation leads to the con- 
 clusion that the problem of attrition in numerator/denominator bundle 
 systems (i.e. the decrease in the absolute value of active wires upon 
 multiplication) can be mapped onto the broken wire problem, i.e. that a 
 "Restorer" can be built which is quite similar to a "Repeater." In both 
 cases tri-state logic is used in the relevant circuits. 
 
TABLE OF CONTENTS 
 
 Page 
 I. INTRODUCTION 1 
 
 1.1 Introduction ---------------_____ i 
 
 1.2 BURP - A Definition of the Problem 2 
 
 1.2.1 The Bundle Repeater Problem 2 
 
 1.2.2 The Bundle Restorer Problem - 3 
 
 II. THE BUNDLE REPEATER 5 
 
 2.1 The Repeaters 5 
 
 2.2 The Formation of Pairs ---- ____ ___io 
 
 2.3 Two Ways of Forming Pairs ------------- ±k 
 
 III. THE BUNDLE RESTORER 18 
 
 3.1 The Restorer 18 
 
 3-2 Preprocessing of the Bundles for the 
 
 Restorer Action ------------------19 
 
 3.3 Two Ways of Achieving a 2 to 1 Reduction of Wires - 20 
 3.1+ Restoration of Inactive Wires by Repeater Stages - - 26 
 
 IV. CONCLUSION 28 
 
 V. APPENDIX 30 
 
 Appendix 1 The BURP System 30 
 
 Appendix 2 The Input System --------------- 3U 
 
 Appendix 3 The Repeater Stages --------------39 
 
 Appendix k The Output Display System -----------1+2 
 
 LIST OF REFERENCES *+5 
 
VI 
 
 LIST OF FIGURES 
 
 Figure Page 
 
 1. Effect of a Hazardous Channel ------------- 6 
 
 2. Effect of Three Hazardous Channels 7 
 
 3. A Remapping of the Wires in the Bundle into 
 
 Three Distinct States ----------------- 9 
 
 k. The Repeater Operation ---------------- n 
 
 5. An r-Repeater System ----------------- 13 
 
 6a. Method 1 of Forming Pairs ---------------15 
 
 6b. Method 2 of Forming Pairs ---------------16 
 
 7. Method 1 for 2 to 1 Wire Reduction -21 
 
 8. Method 2 for 2 to 1 Wire Reduction 2h 
 
 9. Switching Circuits A and B -------------- 25 
 
 10. A Solution to the 'Attrition' Problem 27 
 
 11. System Block Diagram ----------------- 31 
 
 12. Block Diagram of Analog-to-Bundle Converter ------ 35 
 
 13. BCD Display Generator 37 
 
 lU. The Repeater Circuit _________ i+o 
 
 15. Block Diagram of the Output Display System ------ U3 
 

 
I . INTRODUCTION 
 
 1.1 Introduction 
 
 1+ 
 In the time-stochastic machines proposed by Professor W. J. Poppel- 
 
 baum, numbers are expressed in the form of random or pseudo-random pulse 
 
 sequences: Their magnitude is represented by the probability that a given 
 
 time-slot contain a pulse. It has been shown that by utilizing such a 
 
 system of number representation, arithmetic operations can be implemented 
 
 by the use of very simple boolean functions (in the simplest case, the 
 
 product of two numbers is formed by their logical AND) and using some of 
 
 the basic rules of probability. 
 
 In an effort to study the possibility of implementing fail-safe or 
 
 fail-soft machines using probabilistic representations, the use of bundles 
 
 of wires each carrying the numerical information as an average has been 
 
 13 5 
 proposed. ' ' In such a system, the numerical value or 'weight' of a 
 
 bundle is represented by the fraction of the wires in the bundle that is 
 in the 'ON' state at that particular instant. When the weight of the 
 bundle is averaged over a sufficient period of time (determined by the 
 accuracy deemed necessary) the average is equal to the normalized number 
 that the bundle transmits. Note that in many respects the bundle is the 
 space-analog of a finite sequence of time-slots. 
 
 By utilizing the redundancy (multiple wires!) property of the bun- 
 dle, and the time averaging of the wires in the bundle, a fail-soft system 
 can be realized. Such a bundle can be operated in a hostile environment, 
 so that even if a portion of the wires in the bundles is damaged, meaning- 
 ful (although somewhat degraded) information can be salvaged from the 
 wires that are left intact. Furthermore, by averaging the values (or 
 
weight) of the damaged bundle over a sufficient period of time, more 
 
 accurate information can be obtained. 
 
 3 5 
 It has been shown by previous investigations ' that by using a 
 
 system of two bundles to represent a rational number — one bundle represent- 
 ing the numerator and the other bundle the denominator — arithmetic opera- 
 tions can be performed quite easily, again by using some of the basic rules 
 of probability. 
 
 For a simple example, we can look into the process of taking the 
 reciprocal of a rational number. If such a system of two bundles is used 
 to represent the rational number in question, the process of taking recip- 
 rocals can easily be implemented by interchanging the numerator and denomi- 
 nator bundles. A multiply operation, on the other hand, involves simply 
 the ANDing of the respective numerator bundles and the respective denomina- 
 tor bundles. 
 
 1.2 BURP - A Definition of the Problem 
 1.2.1 The Bundle Repeater Problem 
 
 BURP — the bundle repeater /restorer — is a machine designed to enhance 
 the attractiveness of bundle processing when the latter is used in the 
 transmission of signals and the performance of arithmetic. 
 
 The bundle repeater portion of the BURP project enhances the fail- 
 soft feature of bundle processing: Often, a signal has to be transmitted 
 through a series of noisy or hazardous environments or channels. In such 
 a situation, the fail-softness of a bundle processing system is very 
 attractive, since the degradation of some of the wires in the bundle has 
 only a slight effect on the average value of the entire bundle. Further- 
 more, repeater stations can be placed in between the noisy channels. 
 These repeater stations examine the damaged incoming signal bundles for 
 
broken or shorted wires and recondition these unreliable wires in such a 
 way so that the bundle exiting from the repeater station bears a closer 
 resemblance to the original signal. 
 
 By locating these repeater stages in between each hazardous chan- 
 nel, so that each damaged bundle is reconditioned before it is allowed 
 to enter a second hazardous channel, the system is given a further boost 
 in its ability to withstand external damage. This is because the output 
 of the repeater contains more signal carrying wires than its input bundle 
 and consequently the signal bundle at the exiting end of the channel 
 would have a larger number of good wires than it would otherwise have. 
 
 1.2.2 The Bundle Restorer Problem 
 
 It has been demonstrated by previous investigations that one of 
 the problems involved in doing bundle arithmetic — that is, using a numera- 
 tor and a denominator bundle — is the attrition problem : This is the prob- 
 lem encountered when, after successive arithmetic operations , the signal 
 bundles contain less and less active wires, although their ratios still 
 represent the desired result. This means that the resultant bundles would 
 be more susceptible to errors due to damaged wires; and because of the 
 small number of wires in the numerator and denominator bundles, a loss of 
 significance in subsequent arithmetic operations. 
 
 Interestingly, the repeater stages described earlier in the section 
 can be used to alleviate such an attrition problem. By a mapping opera- 
 tion during which the numerator and denominator bundles are degenerated 
 into a single bundle and by passing this bundle through the repeater sta- 
 tions, a resultant bundle can be obtained so that the ratio of those wires 
 in the '+1' state to those in the '-1' state are the same as before, while 
 the total number of the '+!' and '-!' wires are greatly increased. 
 
Subsequently, by reversing the mapping operation, the numerator and 
 denominator bundles can be regenerated. 
 
II. THE BUNDLE REPEATER 
 
 2.1 The Repeaters 
 
 As described earlier, a signal carrying bundle passing through a 
 hazardous environment would produce a resultant "bundle whose signal is 
 not as reliable as that of the original bundle. Let us first look at the 
 case of an N wire bundle. If a number n is to be transmitted, — of the 
 
 wires in the bundle, chosen at random, would be turned 'ON' , while the 
 
 N - n 
 remaining — - — wires would be left in the 'OFF' state. 
 
 If such a system of n active wires are to pass through a risky 
 channel, so that the probability of any wire in the system being made in- 
 active is p, the number of active wires of the resultant bundle at the 
 receiving end would be less than the original number n (see Figure l): In 
 this case, the number of active wires in the bundle would be n(l-p). If 
 subsequent to this, the same bundle is passed through (m-l) consecutive 
 noisy channels with the same risk factor p, at receiving end of the system 
 the total number of active wires at the receiving end would be n(l-p) 
 (see Figure 2). 
 
 The extent of the problem is revealed if numerical values of p and 
 m are considered. If we assume that the risk factor of each channel is 
 0.1 — which means that one out of every ten wires in the bundle would be 
 damaged by passage through the channel — and if there are altogether 6 
 cascaded channels, the percentage of the original n active wires still 
 reliable after passing through these channels would be 
 
 n(l-p) m - n(l-O.l) 6 = 0.5^n 
 This means that a mere 5k% of the original n wires would still be 
 active. This is hardly encouraging because in practice N is finite and 
 

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 is usually of the order of 100. A transmission through such risky chan- 
 nels would degrade the accuracy and significance of the transmitted signal 
 to such an extent that the system would be unacceptable. 
 
 In order to overcome the deficiency of the system described so far, 
 a scheme has been devised whereby the unreliable wires in the bundle are 
 reassigned values based on the information carried by the reliable wires . 
 In effect, these unreliable wires are re-activated again so that the resul- 
 tant number of active wires approximate that of the original bundle enter- 
 ing the risky channels. The scheme utilizes the fact that the active wires 
 are randomly distributed in the bundle of N wires; and it also introduces a 
 third logic level into the system to differentiate the wires that are 
 disturbed or de-activated by the hazardous channel from those transmitting 
 useful information, i.e. those in the '+1' or '-1' state . The third level 
 serves as an indicator of the unreliability of any particular wire in the 
 system, whereas the '+1' and ' -1' states represent the signal levels of 
 the system (see Figure 3). 
 
 A black box called a "Repeater" is used to implement the scheme. 
 After passage through a hazardous channel, the bundle of signal- carrying 
 wires is connected to the Repeater for processing. The Repeater singles 
 out all the damaged wires (that is, the wires in the '0' state) in the 
 bundle and randomly pairs up each of these with one of the remaining 
 reliable wires. Since the wires in the bundle are randomly distributed 
 (a necessary condition for the wires of the system) the probability that 
 an unreliable wire is paired with a wire in the '+1' is proportional to 
 the fraction of 'ON 1 wires in the original bundle: The unreliable wire 
 is then assigned the value of its paired partner: The Repeater, in effect, 
 restores the inactive wires in the system to the active states ('1' or 
 '-!') in proportion to the number of wires that were originally in the 
 

 
 
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 2.2 The Formation of Pairs 
 
 In the previous section, it was mentioned that the unreliable wires 
 in the bundle are detected by the Repeater, so that they can be assigned 
 values based on the information carried in the active wires. In an actual 
 system, however, this is not realistic because it would involve a two-step 
 operation: (a) the search for the unreliable wires and (b) the reassign- 
 ment of new values to these unreliable wires. Such a two-step process is 
 not acceptable because it would require an intimate physical relationship 
 between all the wires, possibly by some form of cross-bar switching: It 
 cannot be determined before-hand which of the wires in the bundle would be 
 damaged while passing through the hazardous channel. 
 
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 is desirable. One approach is to randomly partition the bundle of N 
 wires into N/b sub-bundles, with each sub-bundle having b wires. The b 
 wires in the sub-bundle are connected in such a way that — if a wire in the 
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 From this we can see that the probability of failure of the re- 
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 constant p, which would mean that the larger the value of b, the more 
 reliable the system would be. However, a more detailed study of the 
 problem shows that this is not, in fact, the case. What is more important: 
 There are circumstances when a larger b is even undesirable! 
 
 When b=N, for example, the system degenerates to the case mentioned 
 in the beginning of this section. If all the wires but one in the sub- 
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 This effect is particularly pronounced in the case when b is a large num- 
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 number of active wires in the resultant bundle, the signal that the bundle 
 conveys would be less meaningful and accurate than that of the degraded 
 bundle. In view of this, it would seem that better system performance 
 can be attained by limiting the repeater action of a particular wire to 
 that of its immediate neighbor. That is, if a wire is unreliable , then 
 it should be assigned the value of the adjacent wire . If the latter hap- 
 pens to be unreliable too, then the repeater action is allowed to lapse — 
 the wires are left in their original unreliable state. (in this case, the 
 value of b would be 2 and the probability of failure of the particular un- 
 reliable wires would be p ) . This would not prove to be too much of a 
 handicap if we examine the case of a r-repeater system, i.e. a system con- 
 sisting of r repeaters, as shown in Figure 5. In this case, we can see 
 that the probability of the system failing to compensate for a particular 
 wire (broken previous to entering repeater l) is: 
 
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 2.3 Two Ways of P'orming Pairs 
 
 In the previous section, the way of implementing the repeater was 
 described: If a wire is broken, it is assigned the value of the wire 
 adjacent to it. (This assumes that the wires in the bundle are distributed 
 randomly). Two different methods can be used to achieve the assignment. 
 
 In the first method, if a wire is broken, it is assigned the value 
 of the wire adjacent to it. The "adjacent" wire, however, (which might 
 actually be reliable ) , will not be able to affect the initial wire in any 
 way , even if it is found to be unreliable. This is shown in Figure 6(a), 
 which also includes the truth table of the operation necessary to imple- 
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 An alternative is to group the N wires in the bundle into N/2 pairs. 
 If a particular wire of a pair is found to be unreliable, it is assigned 
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 If both of these wires are unreliable, then the repeater action has failed 
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 A more detailed analysis shows that the likelihoods of the two 
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 in the bundle are randomly distributed, it would be equally likely for any 
 wire in the bundle to be broken during the passage through the hazardous 
 channel. The merit of the first method is that it can be used for any N 
 wire system, be N even or odd. The second system, however, can only be 
 used in systems where N is a multiple of the wires in the sub-bundle 
 (which is 2 in this case), because it relies on the forming of sub-bundles. 
 On the other hand, the second method merits consideration because of its 
 modularity — for any pair of wires in a sub-bundle, all repeater action is 
 conveniently confined to the pair alone. 
 
18 
 
 III. THE BUNDLE RESTORER 
 
 3.1 The Restorer 
 
 As described in previous investigations, one of the most attractive 
 features of a bundle transmission/processing system is the ease with which 
 arithmetic operations can be implemented. In the case when a rational 
 number is represented by the ratio of the 'ON' wires in two bundles, all 
 arithmetic operations (=, -, x, :) can be achieved by simple logical opera- 
 tions involving the numerator and denominator bundles. Consequently, rela- 
 tively complex operations can be obtained with a relatively small number 
 of gate delays. 
 
 A problem associated with such a system is that after repeated 
 operations the number of 'ON' wires in the bundle tend to diminish. This 
 is expected because all such arithmetic operations involve the multiplica- 
 tion of probabilities, i.e. numbers less than one. This is what is known 
 as the 'attrition' problem described by previous investigators. The 
 effect of such an 'attrition' problem is that as more and more operations 
 are performed on these bundles, the number of active diminishes and con- 
 sequently, the accuracy and precision of the arithmetic suffers. This, 
 of course, is in no way different from what happens in fixed point arithe- 
 metic in a digital computer! 
 
 During the study of the BURP system, it was noticed that a varia- 
 tion of the Repeater system will also solve the 'attrition' problem very 
 nicely, utilizing to this effect the third logic level representing the 
 unreliable state. In the Restorer system, a transformation is done on 
 the wires in the numerator and denominator bundles so that the inactive 
 wires are mapped onto the '0' state of the repeater system. In other 
 words, an inactive wire in the Restorer system is considered to be 
 
19 
 
 unreliable, and through the Repeater stages, a fraction of these inactive 
 wires are restored by the repeater action. In effect, the Restorer re- 
 stores the accuracy and precision of the bundle system. 
 
 3.2 Preprocessing of the Bundles for the Restorer Action 
 
 Let us first consider a bundle system of number representation 
 where a number is represented by the ratio of active wires in a numerator 
 and a denominator bundle, each of which have N wires. If, for example, 
 N/1+ wires in the numerator bundle are active and if N/8 wires in the de- 
 nominator are active, the ratio of the active wires in the two bundles, 
 namely 
 
 HA „ 
 
 N/8 " d 
 
 gives the value of the number we want the bundles to represent. 
 
 In such a system, the first step in the preprocessing is to map 
 the active wires in the numerator bundle onto a '+1' logic level and to 
 map the active wires in the denominator bundle onto the '-1' logic level . 
 The inactive wires in the two bundles can then be mapped onto the '0' level, 
 used previously as the 'unreliable' level. This results in a bundle con- 
 sisting of 2N wires, where the ratio of the wires in the '+1' state and 
 those in the '-1' state represents the number that was previously repre- 
 sented by the two bundle system. If we now take a_ 2N-wire Repeater sys- 
 tem , the number of active wires in the resultant 2N wire bundle can be 
 increased by the repeater action described before, so that at the output 
 of the Repeater stages the total number of active wires (wires in the '+1' 
 and '-1' states) is increased, while the ratio ('+l'/'-l') remains the 
 same. 
 
20 
 
 A problem arises, however, when each of the numerator and denomina- 
 tor bundles has N wires , while we have a repeater system that has only N 
 wires: A way has to be found to reduce the 2N wires of the two bundles 
 down to a single bundle of N wires. 
 
 This can easily be done in the case when the total number of active 
 wires in the two bundles is less than N, as described earlier on in the 
 section. We can seek out all the active wires in the numerator and denomi- 
 nator bundles and assign them the value of '+1' and ' -1 ' respectively, 
 while the remaining wires in the N-wire bundle can be left in the '0' 
 state. We can then have a nicely transformed bundle suitable for level 
 restoration for the Repeater stages. 
 
 The problem is more severe when the total number of active wires 
 in both bundles is larger than N and when the Repeater stages can only 
 handle N wires. In this case, a way has to be found to produce a 2 to 1 
 reduction of wires. 
 
 3- 3 Two Ways of Achieving a 2 to 1 Reduction of Wires 
 
 Method 1 
 
 The simplest method for achieving a 2 to 1 reduction of wires is to 
 randomly select N/2 wires from each of the numerator and denominator bun- 
 dles and group them together into a single bundle of N wires. The active 
 wires that are in the first group are then assigned values of '+1' while 
 those active wires that are in the second group, namely those selected 
 from the denominator bundle, are assigned values of '-1'. The inactive 
 wires from both groups are assigned the '0' state. If these two groups 
 of wires are then collected into a single bundle of N wires, we have 
 achieved a 2 to 1 reduction of wires (Figure 7). 
 
 We can visualize this very well if we assume that n of the wires 
 of the numerator bundle are active while n of those in the denominator 
 
21 
 
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 UJ 
 
 UJ 
 
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 uj a z 
 
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22 
 
 bundle carry signals. In this case, the number represented by the two 
 bundles is: 
 
 n 1 /N n n 
 
 n 2 /N n 2 
 
 If N/2 wires are randomly selected from the numerator bundle, it is 
 
 likely that the number of active wires selected will be approximately 
 
 n ,/2. Similarly if N/2 wires are randomly selected from the denominator 
 
 bundle, the number of active wires selected will be approximately n /2. If 
 
 these randomly selected wires are combined into a single bundle of N wires, 
 
 the number of '+1' wires will be n /2 while the '-1' wires will be n p /2. 
 
 Such a bundle will represent the number: 
 
 n 1 /2 n ± 
 n 2 /2 n 2 
 
 i.e. the number represented by the original numerator and denominator bun- 
 dles. Such a bundle of N wires can then be passed onto the Repeater stages 
 for restoring the inactive wires. 
 
 This method, however, is not really ideal because we can easily see 
 that statistically el /2 of the active wires from the numerator bundle and 
 n_/2 active wires from the denominator bundle are discarded in the wire 
 reduction operation. Since we start out with a relatively small number 
 of active wires in the first place, the resultant N wire bundle has much 
 fewer active wires from each of the bundles for forming the rational 
 number n /n : Such a method of wire reduction will invariably introduce 
 a substantial quantity of error, which probably cannot be tolerated by a 
 system that has few active wires anyway. 
 
23 
 
 Method 2 
 
 This second method for the reduction of wires is similar in prin- 
 ciple to that of the first . Just as before , "both the numerator and de- 
 nominator bundles are each randomly divided into two bundles of N/2 each. 
 Each of the two sub-bundles of N/2 wires of the numerator bundle is 
 paired randomly with a sub-bundle of the denominator bundle, as shown in 
 Figure 8. These two groups of N wires are passed onto a switching net- 
 work that performs the function represented by the truth table of Figure 9. 
 
 Let us say that the numerator bundle is divided into two sub-bundles, 
 c and c p ; while the denominator bundle is divided into two sub-bundles, 
 c_ and Ci . Let us also assume that c. is paired with c_ and that this 
 pair of sub-bundles is connected to switching network A. Similarly, c 
 and c< are paired together and are connected to switching network B, as 
 shown. Each of the N/2 wires of c. is randomly paired with another wire 
 from c„ so that N/2 pairs of wires are formed. Each of these pairs is 
 considered by the switching circuit A, which performs the operation in 
 Figure 9(a), and by doing so a two to one reduction of wires is obtained 
 for c, and c : In effect, switching circuit A outputs a single wire for the 
 incoming pair. 
 
 A study of the function performed by switching circuit A reveals 
 that the values of the wires in c are dominant over those of c . When- 
 ever there is signal in a wire from c , switching circuit A automatically 
 outputs a wire of value '+1'. The value of the wire from c is only con- 
 sidered when the corresponding wire from c is inactive, in which case, 
 switching circuit A outputs a wire of value '-1*. In the case where both 
 the wires from c and c are inactive, a wire with value '0' results. 
 
 A similar observation can be made about c and c, , but in this 
 case Ci is more favorably biased, and switching circuit B outputs a wire 
 
2k 
 
 UJ 
 
 c 
 o 
 
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 -p 
 
 O 
 
 a; 
 
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 en 
 
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 ^ tr 
 
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 o 
 
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25 
 
 SWITCHING CIRCUIT A 
 
 
 WIRE 
 FROM Ci 
 
 +1 
 
 WIRE +1 
 FROM 
 
 Cs 
 
 +1 -1 
 
 ♦ 1 
 
 SWITCHING CIRCUIT B 
 
 WIRE 
 FROM C 2 
 
 
 +1 
 
 
 
 WIRE +| 
 
 -1 
 
 -1 
 
 FROM 
 
 
 
 c « 
 
 +1 
 
 
 
 Figure 9. Svitching Circuits A and B 
 
26 
 
 of value '-1' whenever the wire from c< is active, while the wire from c p 
 is considered only when the corresponding wire from c, is inactive. 
 
 This is very similar to the reduction procedure described in 
 Method 1. The only difference is that the other half of the wires that 
 are usually discarded are now retained and that they do affect the com- 
 position of the resultant N wire bundle . This is achieved automatically 
 by pairing the appropriate sub-bundles and by using the peculiar switching 
 function of the switching networks. 
 
 Statistically, this method will yield more active wires for con- 
 sideration by succeeding repeater stages and chances are that a more 
 accurate representation of the original n /n~ number can be attained. 
 
 3.U Restoration of Inactive Wires by Repeater Stages 
 
 As mentioned earlier, the resultant bundles of N wires are pro- 
 cessed by the succeeding Repeater stages so as to generate a larger num- 
 ber of active wires. Since, in the resultant bundle, the ratio of the 
 '+1' wires to that of the '-1' wires approximate the ratio of active 
 wires in the numerator and denominator bundles, such a bundle used at the 
 input of the Repeater stages will indeed generate an output bundle that 
 still maintains the n /n~ ratio, while greatly increases the number of 
 '+1' and '-1' wires at the output . This shows then that by the use of a 
 system as shown in Figure 10, a solution to the 'attrition' problem can 
 be realized. 
 
27 
 
 !<j 
 
 
 j> 
 
 r-REPEATER 
 STAGES 
 
 
 K 
 
 , 
 
 :> 
 
 a 
 
 H 
 
 ^> 
 O 
 
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 -P 
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 p 
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 < 
 
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28 
 
 IV. CONCLUSION 
 
 A system that restores the number of active wires as well as 
 solving the 'attrition' problem of a bundle transmission/processing system 
 has been described in the preceding pages. The restoration of active 
 wires enhances the attractiveness of a bundle/transmission system since 
 the ability of such a system to withstand damaged wires as information 
 passes through repeated hazardous channels is greatly increased. On the 
 other hand, the solution of the 'attrition' problem enhances the attrac- 
 tiveness of using bundles in the processing of arithmetic, since in addi- 
 tion to the simplicity of arithmetic, the accuracy of computation can be 
 maintained throughout succeeding levels of computation. 
 
 A failure in the Repeater or Restorer circuits, however, may cause 
 errors. The probability of this occurring is not very high, because 
 the Repeaters and Restorers are located in relatively 'safe' environm- 
 ments ; and, what is more, the effect of a damaged Repeater circuit is 
 not severe because of the modularity of the Repeater /Restorer stages — 
 any failure inside a Repeater circuit affects the value of at most one 
 wire . Consequently, the performance of BURP is, in the worst case, no 
 worse than that of a comparable system without the repeater action; 
 and, under most circumstances, a Repeater /Restorer system is more reli- 
 able , because the probability of system failure is minor compared to 
 the probability of wire damage caused by a hazardous channel. 
 
 In the appendices that follow, we shall describe a bundle repeater/ 
 restorer system (called BURP) that has actually been constructed to demon- 
 strate the feasibility of such a system. Not surprisingly, the system 
 shows, when it is used to demonstrate the repeater action, a great ability 
 
29 
 
 to withstand damage sustained during passage through the channels separat- 
 ing the Repeater stages. When it is used to demonstrate the restorer 
 action, the number of active wires in a bundle can be restored close to 
 % after two or three Repeater stages. 
 
30 
 
 V. APPENDIX 
 
 Appendix 1 
 The BURP System 
 
 This section describes a 5-stage Repeater system that has been con- 
 structed to demonstrate the feasibility of the BURP system. A block dia- 
 gram is shown in Figure 11. 
 
 BURP is essentially divided into two systems: The Repeater portion 
 that can be used to demonstrate the repeater action (namely the ability of 
 the repeater stages to compensate for damaged wires), and the Restorer 
 portion that can be used in conjunction with the Repeater stages to demon- 
 strate the resoration of active wires in a two-bundle processing system. . 
 As a result, the Repeater and Restorer portions of the machine are not very 
 well separated physically, although the principles that they demonstrate 
 are distinct. 
 
 Each of the Repeater stages can handle a maximum of 6U wires, which 
 essentially means that the numerator and denominator bundles of the re- 
 storer section have to be made up of 6k wires each too. Each of the outputs 
 of a Repeater stage is connected at random by a wire to an input circuit of 
 the next repeater stage, and each of these connections can be severed and 
 reconnected at will to demonstrate the effects that a hazardous channel 
 has on the system. 
 
 When a demonstration of the repeater action is desired, an analog 
 input is converted into a bundle representation, i.e. the number of active 
 wires is proportional to the analog input. These 6h wires generated by 
 the converter are then passed onto the 5 Repeater stages that follow. 
 
 Hazardous channels of different risk factors can then be simulated 
 by severing the inter-connecting wires between Repeater stages; a risk 
 
31 
 
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 en 
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 Q- 
 
32 
 
 factor of .5 can be had, for example, by randomly disconnecting 32 of 6U 
 wires between two Repeater stages. After the fifth and last Repeater 
 stage, the active wires are summed and can be compared to the original 
 analog number to demonstrate the effects of the intervening repeaters. 
 Each of these Repeater stages can also be disabled if desired to demon- 
 strate the effects of one or all of the Repeaters on the quality of the 
 received signal. 
 
 If a demonstration of the Restorer action is desired, then two analog 
 numbers representing the values of the numerator and denominator bundles 
 are converted into bundle notation and, after the 2 to 1 wire reduction 
 procedure of the Restorer stage, the resultant 6k wires are then passed 
 onto the Repeater stages. Again, the output of the last Repeater stage 
 is monitored to show the total number of active wires and the composition 
 of the regenerated bundles. 
 
 To facilitate comparision, the 6h wires of the system are mapped 
 onto a range of 0.00 to 1.00 by a bundle-to-BCD converter that displays 
 the analog input in 2 1/2 BCD digits. Similarly, an output display genera- 
 tor displays the desired function of the output bundle from the last Re- 
 peater stage in BCD form. So instead of showing the total number of active 
 wires in the appropriate bundles, the input and output displays show the 
 percentage of the wires that are active; however, since a 6U-wire system 
 has an accuracy of less than 1% the output displays will show numbers that 
 are more precise than the bundles can actually demonstrate. 
 
 Another word about the output display is necessary. Since the same 
 output display serves both the Repeater and the Restorer functions, it 
 must be versatile enough to interpret the contents of the output bundles 
 and be able to display its interpretations in meaningful forms. The 
 output display can generate the following functions: a) Z'+l', b) E'-l*, 
 
33 
 
 c) Z'+l'/E'-l', and d) E'+l'/U'+l 1 + E'-l')- As will "be explained 
 later, a), b) and d) are helpful in interpreting the resultant output bun- 
 dles in a demonstration of the Repeater action, while a), b) and c) are 
 helpful in the case of the Restorer. 
 
3U 
 
 Appendix 2 
 The Input System 
 
 The input system converts a number represented by an analog voltage 
 into a form that the system can operate on. The analog number is mapped 
 onto a bundle of 6k wires so that the number is represented by the ratio 
 of the '+1' wires in the bundle to the total number of wires in the bundle; 
 namely 6k. To facilitate the demonstration of the system, this analog 
 number is also represented as a 2 1/2 digit BCD number, and this latter 
 is displayed on the machine. 
 
 Figure 12 shows a block diagram of the analog-to-bundle converter 
 that maps the analog number onto the 6k wires. The basic part of this 
 sub-system is a 7-bit tracking analog-to-digital converter that has a 
 maximum digital range restricted to a count of 2 (i.e. 6k). This means 
 that the maximum number of analog numbers that can be represented is 61+ 
 and these numbers are mapped onto the range of 0000000 to 1000000. 
 
 The 7-bit output of the A/D converter is used to drive the 6U wires 
 in the bundle. The least significant bit, 2 , for example, is used to 
 drive one wire only, while the 2 bit is used to drive 2 wires, the 2 bit 
 dirves 32 wires. The 2 bit is allowed to drive only 1 wire, however when 
 a 1000000 count is noted, the rest of the wire drivers are enabled so that 
 all 6k wires of the system are turned on. The wire drivers are designed 
 so that if a particular bit of the A/D converter is 'ON' , the wire driver 
 has a '+1' level output. Otherwise the output assumes a level of '-1'. 
 
 As indicated on the system block diagram of Figure 11, two 7-bit 
 latches are used to store the A/D converter outputs when the system is 
 used in the Restorer mode. In this case, one 7-bit latch would store the 
 
35 
 
 in 
 Ul 
 
 K 
 
 <0 
 
 u 
 
 <D 
 -P 
 
 U 
 
 > 
 
 a 
 o 
 o 
 
 Q 
 UJ 
 
 CO 
 
 UJ 
 
 > 
 
 £ 
 
 o 
 
 UJ 
 
 cc 
 
 UJ 
 
 > 
 
 CC 
 
 £ 
 
 CVJ 
 
 o 
 
 *4 
 
 M K> 
 
 * « 
 
 <0 
 
 CM 
 
 CM 
 
 2 
 
 7-BIT 
 TRACKING 2 
 
 CM CM 
 
 BE 
 
 UJ 
 
 o tr 
 
 "v UJ 
 < > 
 
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 o 
 
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 u 
 
 3 
 
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36 
 
 numerator number while the second would store the converted denominator 
 number. These two numbers are entered into the input system through the 
 same potentiometer and A/D converter, and so have to be entered sequentially 
 and stored separately in the two latches. When the Restorer mode of opera- 
 tion is desired, these two 7 -bit numbers are passed onto the Restorers, 
 which reduce the 2 x 6h wires to a single bundle of 6k wires and at the 
 same time, serve as wire drivers for the bundle that goes onto the Repeater 
 stages. 
 
 Also indicated in Appendix 1 is the added function of the input sys- 
 tem to display the analog entries to the BURP system in BCD form. In the 
 repeater mode, the analog input is connected directly to the BCD generator 
 and so the conversion of the to 6h bundle notation to the 0.00 to 1.00 BCD 
 notation is accomplished on-line. On the other hand, this on-line opera- 
 tion of the BCD generator is not possible for both of the bundles when 
 the restorer mode of operation is required. In this situation, the analog 
 number for one bundle is converted first into the bundle notation and the 
 BCD generator regenerates the BCD representation from the latter. The 
 resultant 2 1/2 digit BCD number is stored in another latch. A similar 
 conversion is done for the second bundle and the BCD result is similarly 
 stored in a 7-bit latch. 
 
 The generation of the BCD numbers from the bundle notation is a 
 relatively simple process. A block diagram of this sub-system is shown in 
 Figure 13. Whenever the system senses that a new number has been entered 
 
 through the input potentiometer, a signal starts a clock generator that 
 
 2 
 feeds into the 10 counter and into a binary rate multiplier. The latter 
 
 is a sub-system that allows n/6h of the pulses entering it to exit at its 
 
 output. Therefore, if n is set to be equal to the 7-bit output of the A/B 
 
37 
 
 
 
 
 
 
 1 
 
 
 
 w 
 
 
 
 + 
 
 
 
 
 
 1 
 
 + 
 
 
 w 
 
 w 
 
 m 
 
 _v. 
 
 "V 
 
 w» 
 
 ^H 
 
 •-4 
 
 1 
 
 + 
 
 + 
 
 w 
 
 w 
 
 w 
 
 o 
 -p 
 a) 
 
 Jh 
 
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 a; 
 
 * 
 H 
 
 P< 
 w 
 
 ■H 
 
 Q 
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 pq 
 
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 0) 
 
 no 
 
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38 
 
 converter and if the pulses exiting from the binary rate multiplier are 
 counted by a 2 1/2 digit BCD counter a conversion of the bundle notation 
 
 to the BCD notation can be achieved at the output of the BCD counters. 
 
 2 
 The clock generator is disabled whenever the 10 counter overflows. As 
 
 noted earlier, this BCD number can then be stored in latches and displayed 
 
 when needed. 
 
39 
 
 Appendix 3 
 The Repeater Stages 
 
 In Chapter 2, two methods were shown for the assignment of pairs 
 for the Repeater stages. Method 2, as shown in Figure 6(h) was used in 
 the design. Figure ik shows the actual circuit that was used in the BURP 
 system to implement the required function. 
 
 Let us first look at the case when the Repeater stage is in the 
 'OFF' position, i.e. if the ON/OFF line is kept in the grounded state. 
 This means that resistors R and R become the grounded loads for outputs 
 A 1 and B'. In this case, when input A is in the '+1' state, Q would be 
 'OFF*, while Q would be 'ON'. This means that Q would turn 'ON' and 
 Q, would turn 'OFF' , and the output would assume a '+1' state. 
 
 If input A is in the '-1' state, then Q and Q, would be 'ON' 
 while Q and Q would be 'OFF' , resulting in a ' -l f state at the output 
 A'. 
 
 If, on the other hand, input A is either grounded or left open, 
 then both Q and Q would turn on and consequently Q and Q, would be off. 
 This means that output A' would be left in the open or high-imepdence state, 
 
 It is now evident that if the ON/OFF line is allowed to float, out- 
 puts A; and B; would follow the inputs A and B respectively when their re- 
 spective inputs are either in the '+1 1 or '-1' states. If input A is 
 grounded for example, Q and Q« would be OFF, and so A 1 would follow the 
 logic level of B ! , since A 1 and B' are connected by resistors R and R . 
 This means that wire A (since it is an unreliable wire) is assigned the 
 value of its neighbor B, and hence the repeater action is accomplished. 
 In the event that both A and B are grounded or are in the 'OPEN' states, 
 
ko 
 
 ON/OFF O 
 
 B 
 "0" O- 
 
 m ^. 
 
 V+ 
 
 A 
 
 
 *"' 
 
 
 _^ II , II ll/»H - _ II M 
 
 ■^ + , , OR - 
 
 OR - 
 
 Figure lU. The Repeater Circuit 
 
kl 
 
 A' and B' would both assume the high-impedence state '0', and the repeater 
 action is assumed to have failed. 
 
 Thirty-two of these basic cells are used in each of the Repeater 
 stages. In a 5-Repeater system such as BURP, a total of l60 of such 
 repeater cells is utilized. 
 
U2 
 
 Appendix h 
 The Output Display System 
 
 It was mentioned in Appendix 1 that the output display system is 
 capable of generating the following functions: a) E'+l', b) Z'-l', 
 c) Z'+l'/E'-l' , and d) E'+l'/U'+l* + Z'-l* ). 
 
 In the repeater mode, an analog number is represented by the frac- 
 tion of the wires in the bundle that are in the '+1' state, compared to 
 the total number of '+1' and '-l 1 wires. Therefore, E'+1'/(Z*+1' + Z'-l*) 
 will tell us what the original number has degenerated into after passage 
 through the 5 hazardous channels. E'+l 1 and Z'-l', on the other hand, 
 will tell us what fraction of the 6k wires in the output bundle are in 
 the active state and hence give an indication of how accurate the output 
 number actually is. 
 
 In the restorer mode, an analog number is represented by the ratio 
 of the number of ' +1' wires to the number of ' -1 ' wires. Therefore, 
 E'+l'/E'-l' will show what this number has become after the 5 stages of 
 restoring action. The E'+l' and E'-l 1 displays are more important in this 
 restorer mode of operation, however, because they tell us the percentage 
 of the wires in the output bundle that are active, and the main function 
 of the Restorer is to restore the number of active wires. 
 
 A block diagram of this system is shown in Figure 15- The inputs 
 to the output display generator are driven by the last Repeater stage. 
 Each of the 6U wires of the bundle, when it is in the '+1' or '-1' state, 
 act as a current source or current sink respectively for the '+1' or '-1' 
 summers. When a wire is in the '0' state, however, it is neither a current 
 source nor a current sink. The '+!' and '-!' summers will give a relative 
 
k3 
 
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 indication of the percentage of the '+1 1 and '-1' wires in the output 
 bundle. These two current sums are then processed by an analog adder 
 and an analog divider to generate the Z'+V/Z'-V and I'+l'/U'+l' + 
 E'-l 1 ) functions that the system requires. These analog voltages are 
 then selectively fed to an A/D converter and the 2 1/2 digit BCD output 
 is displayed by the machine. 
 
1*5 
 
 LIST OF REFERENCES 
 
 1. Afuso , C. , "Analog Computation with Random Pulse Sequences," Department 
 
 of Computer Science Report No. 255, University of Illinois, Urbana, 
 Illinois, February 1968. 
 
 2. Esch, J. , "RASCEL - A Programmable Analog Computer Based on a Regular 
 
 Array of Stochastic Computing Element Logic," Department of Computer 
 Science Report No. 332, University of Illinois, Urbana, Illinois, 
 June 1969. 
 
 3. Coombes, D. , "SABUMA - Safe Bundle Machine," Department of Computer 
 
 Science Report No. Ul2, University of Illinois, Urbana, Illinois, 
 August 1970. 
 
 km Poppelbaum, W. J., Computer Hardware Theory, Macmillan, New York, 1972. 
 
 5. Ring, D. , "BUM - Bundle Processing Machine," Department of Computer 
 Science Report No. 353, University of Illinois, Urbana, Illinois, 
 October 1969. 
 

 
BLIOGRAPHIC DATA 
 EET 
 
 1. Report No. 
 
 uiucdcs-r-t 1 +-689 
 
 2. 
 
 3. Recipient's Accession No. 
 
 Title and Subtitle 
 
 BURP: A BUNDLE REPEATER AND RESTORER 
 
 5. Report Date 
 
 December 197-^ 
 
 
 6. 
 
 Author(s) 
 
 Bernard Kapang Tse 
 
 8. Performing Organization Rept. 
 
 No - uiucdcs-r-tU-689 
 
 Performing Organization Name and Address 
 
 Department of Computer Science 
 
 10. Project/Task/Work Unit No. 
 
 University of Illinois at Urbana-Champaign 
 Urbana, Illinois 6l801 
 
 11. Contract /Grant No. 
 N000-11+-67-A-0305-002U 
 
 Sponsoring Organization Name and Address 
 
 Office of Naval Research 
 219 South Dearborn Street 
 
 13. Type of Report & Period 
 Covered 
 
 technical 
 
 Chicago, Illinois 6o6oh 
 
 14. 
 
 Supplementary Notes 
 
 Abstracts 
 
 This thesis examines the problem of broken vires in bundles: It is shown that a 
 ich better approximation to the original information can be obtained by connecting the 
 ids of borken wires to an arbitrary neighbor in a so-called "Repeater." Further inves- 
 gation leads to the conclusion that the problem of attrition in numerator/denominator 
 indie systems (i.e. the decrease in the absolute value of active wires upon multiplica- 
 on) can be mapped onto the broken wire problem, i.e. that a "Restorer" can be built 
 dch is quite similar to a "Repeater." In both cases tri-state logic is used in the 
 levant circuits. 
 
 Key Words and Document Analysis. 17a. Descriptors 
 
 me -Stochastic Machines 
 
 .ndom Pulse Sequence 
 
 .il-soft 
 
 indie Representation 
 
 . Identifiers /Open-Ended Terms 
 . COSATI Field/Group 
 
 Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 52 
 
 unlimited distribution 
 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 22. Price 
 
 »M N TIS-35 ( 10-70) 
 
 USCOMM-DC 40329-P7I 
 
* 
 
 
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